Coupling on the MHT - Supplementary material

Transcription

1 Coupling on the MHT - Supplementary material February, S. Map of the GPS network Figure S: Map indicating the names and position of the continuous GPS stations used in this study to determine the pattern of coupling on the MHT. S. Derivation of secular velocities from the GPS time series S.. Noise model for the inversion Several studies of GPS time series have established that the daily estimates of GPS positions are temporally correlated (Langbein and Johnson, 997; Zhang et al., 997; Mao et al., 999;

2 Williams, 3a; Williams et al., 4). Assuming a purely white noise model is therefore incorrect and although it doesn t affect much the value of the final parameters inverted for, it results in a dramatic underestimation of their uncertainties. We thus add to the white noise in our GPS time series a component of colored noise, i.e. a noise that has a power spectrum of the form: P κ (f) f κ, (S) where f is the temporal frequency, and κ is called the spectral index (Mandelbrot and Van Ness, 968). The spectral index is estimated for each time series by first fitting equation () (see main paper) to the time series assuming a white noise, and computing a periodogram of the fit s residuals. The spectral index is then estimated by fitting to the power spectrum a combination of white and colored noise (figure S): P (f) = P + P c f κ, (S) where P and P c are the respective amplitudes of the white and the colored noise. Once the spectral index is estimated, we build the covariance matrix of the data as the sum of white and colored noise covariance matrices. The relative amplitudes of both noises are estimated by a Maximum Likelihood Estimation (MLE) method (Williams et al., 4). The covariance matrix for the white noise is the usual diagonal covariance matrix C w = diag(σ, σ,..., σ n), where σ i is the standard deviation of data point number i. The colored noise covariance matrix C κ is built following an adaptation of the method described in Williams (3a): C κ = t s TT T, where t s is the sampling interval (so t s = day for GPS time series), and the matrix T is defined as: ψ... ψ ψ... T = ψ ψ ψ..., (S4) ψ n ψ n ψ n... ψ where the quantities ψ n are defined by the recurrence formula: { ψ = ψ n+ = n κ/ n+ ψ n. The rows and columns corresponding to times with no data are then removed from the covariance matrix. The final data covariance matrix is given by (S3) C D = a C w + b C κ,. (S)

3 SIMR north! Residuals power spectrum DAMA east! Residuals power spectrum Spectral index:! =!.47 Spectral index:! =!.87! (a) SIMR, north component JMSM north! Residuals power spectrum Spectral index:! =!.93 Spectral index:! =!.6 (c) ODRE, north component (d) JMSM, north component IISC north! Residuals power spectrum CHLM north! Residuals power spectrum f (yr!) f (yr )! Spectral index:! =!.6 Spectral index:! =!.78 (b) DAMA, east component ODRE north! Residuals power spectrum f (yr!) f (yr ) f (yr!) f (yr!) (e) IISC, north component (f) CHLM, north component Figure S: Power spectra of the noise (blue curve) for a sample of time series and fit assuming a combination of white noise and colored noise (red curve). The spectral index κ of the colored noise is indicated on each plot. where a and b are the parameters to be estimated by MLE, measuring respectively the amplitude of white and colored noise. Assuming a Gaussian distribution of the uncertainties on GPS positions, the likelihood that has to be maximized with respect to a and b is then likelihood(cd ) = rt CD r e, (π)n/ (det CD )/ (S6) where r is vector of residuals of the fit and N is the number of daily GPS positions available. The fit and residuals on some time series are shown on figure S3. 3

4 SIMR! North displacement and fit DAMA! East displacement and fit!r = !r =. 998 SIMR! North displacement residuals ODRE! North displacement and fit 8 JMSM! North displacement and fit 6 (b) DAMA, east component!r =.!r = ODRE! North displacement residuals JMSM! North displacement residuals 4 (a) SIMR, north component (c) ODRE, north component (d) JMSM, north component IISC! North displacement and fit CHLM! North displacement and fit 4 3!r =.44!r =.68 6 IISC! North displacement residuals 8 9 CHLM! North displacement residuals DAMA! East displacement residuals (e) IISC, north component (f) CHLM, north component Figure S3: Fits and residuals of the continuous GPS time series at some stations. For each time series, the upper plot shows the raw data (blue curve) and the fit (green curve) with equation (). The value of the reduced chi square χr of each fit is indicated on the upper plot s lower right corner of the corresponding figure. S.. Uncertainties due to unmodeled steps in the time series Steps in the time series can be of many different origins, being actually tectonic, environmental or coming from equipment malfunction, human error, etc. (Williams, 3b). The 4

5 ones large enough to be detected are included in the model (equation ()), but smaller ones remain unnoticed and affect the estimates of model parameters and their uncertainties. Therefore, those uncertainties have to be adjusted accordingly. For convenience, we will assume that those unmodeled steps account for all the errors on the model. Those steps are assumed to happen at a frequency ν, and to have a random Gaussian amplitude N (, σx). The standard deviation on the secular velocity due to those steps is then (Williams, 3b) σ v = σ x ν, (S7) T where T is the length of the time series. In the case of our GPS time series, the amplitude of the steps that were actually detected was always greater than. time the median value of the uncertainties on the daily positions in the time series. We hence take σ x = σ D, where. denotes the median value and σ D is the uncertainty on daily positions of the time series. We estimate ν through the following considerations. First, the steps that were large enough to be detected in the time series happened on average once every years. Assuming that the smaller the steps are, the more frequent they would be, the value for ν should be greater than /. On the other hand, a value of ν overestimated (ν > in this case) results in larger uncertainties on the secular velocity, and eventually leads to values of a reduced chi square smaller than when one fits the Euler pole of the Indian plate in the ITRF reference frame (see section 3.3), indicating that the uncertainties on the GPS velocities are probably overestimated. As a result, we chose a value of ν = /3, which gives the final formula for the uncertainties on the secular velocity due to unmodeled steps: σ v = σ D 3T. (S8) The velocities and corresponding uncertainties that we obtain at the GPS stations used in this paper as well as at the DORIS stations COLA and EVEB are given in table S. S.3 Slip resolution and Laplacian The result of our inversion should be assessed in view of its resolution. This information is contained in the resolution matrix: R = ( G T C d G T + Λ T Λ ) G T C d G T, (S9) where G is the Green s matrix defined in equation () from the main paper, C d is the data covariance matrix and Λ is the Laplacian matrix. The diagonal of R tells how well the slip value on each patch can be retrieved by the inversion. However, it doesn t express how each patch correlates with its neighbors. This information is contained in each of the individual columns of R: column number i is the vector of parameters (i.e. the slip on each patch) returned by the inversion from an input dataset corresponding to a unit slip on patch i and no slip on other patches. Usually, what the inversion returns is slip on a more or less

6 spread area centered on patch i. The characteristic size of this area is estimated by fitting a bell curve to the slip on the patches as a function of distance to patch i (Lohman, 4), and taking the standard deviation of that bell curve. Namely, for each patch i, we find the distance w i that minimizes the quantity: N p ( χ i = j= R ji R ii ) d ij e w i, (S) where N p is the number of patches on the fault, R ji is the value of the coefficient (j, i) of the resolution matrix R (row j and column i), and d ij is the distance between patches i and j. This idea of an estimate of the resolution scale on each patch is also used in order to more efficiently smooth our model by weighting the Laplacian according to the resolution on each patch. Since the Laplacian matrix is not yet available (this is what we try to determine), we compute a first resolution matrix using the Moore-Penrose pseudoinverse matrix (Aster et al., ), keeping only the singular values larger than % of the maximum one. We then compute how far each patch correlates with its neighbors with the method previously described applied to this resolution matrix. Finally, each line of the Laplacian matrix is weighted by the decimal logarithm of the resolution size on the corresponding patch. S.4 Supplementary figures on the pattern of coupling on the MHT S.4. Laplacian smoothing On figure S4 we test how different values of the Laplacian smoothing affect the estimate of the moment deficit accumulated every year. Weights assigned to the Laplacian too small Station ID Site name Latitude ( N) Longitude ( E) Elevation (m) BAN Bangalore DGAR Diego Garcia GUAO Guao GUAM Guam Observatory HYDE Hyderabad IISC Indian Inst. Science KUNM Kunming LHAS Lhasa LHAZ Lhasa POL Poligan IVTAN SELE Selezaschita TAIW Taipei URUM Urumqi WUHN Wuhan Table S: List of IGS sites included in the daily regional processing. 6

7 Velocities in ITRF (mm/yr) Time of operation Station lon ( E) lat ( N) V e V n V u Init. End DAMA ± ± ±.37 Nov 997 current GUMB ± ±.34.7 ±.9 Nov 997 current SIMR ± ±.8.68 ±.7 Nov 997 Apr. BRNG ± ± ± 3. Mar 4 May 9 BRN ± ± ±.48 May 9 current CHLM ± ± ±. Mar 4 current JMSM ± ± ±.36 Ma. 4 current KKN ± ±.4.3 ±.3 Jan 4 current KLDN ± ± ±.7 Apr 4 current MSTG ± ±..79 ± 3.7 Apr 4 Sept 4 a MST ± ±.3.8 ± 4.9 Oct 9 current ODRE ± ±.37. ±. Mar 4 current SIM ± ±..4 ±.8 Mar 4 current SRGK ± ±.9 4. ± 3. Mar Feb 7 TPLJ ±. 3.4 ±.3. ±.3 Mar 4 current BMCL ± ± ±.9 Mar 7 current DLPA ±.63 ±..3 ±.68 May 7 current GRHI ± ± ±.64 May 7 current JMLA ± ±.4. ±.6 May 7 current NPGJ ± ±.74.6 ±.63 May 7 current BYNA ±.38 6 ± ±.39 May 8 current DNGD ± ±.8.48 ±.8 May 8 current DRCL ± ±..64 ±.34 Mar 8 current GNTW ± ±.63.7 ± 3. Apr 8 current RMJT ± ±.48.4 ± 4.36 Oct 8 current RMTE ± ± ±. Sep 8 current SMKT ±.7.84 ± ±.83 May 8 current SYBC ±.8.93 ± ±.88 Oct 8 current CUOM ±.9.38 ±..3 ±. Oct 6 current JRGR ±.9.39 ±. 3.3 ±.7 Mar 7 current XGBA ±. 8. ± ± 4. Mar 7 Sep 7 b YARE ± ± ±.86 Oct 6 current ZHXZ ± ±.4.87 ±.88 Oct 6 current MALD ± ±.49. ±.93 Jul 999 May 6 HYDE ± ± ±.8 Sept current IISC ± ±.3. ±.6 Oct 997 current COLA ± ± ± 3.78 Jan 993 Sep 4 EVEB ± ± ± 3.64 May 993 current Table S: Estimates of the secular velocity at the continuous GPS stations in ITRF and dates of operation of each station. The uncertainties on the velocities indicated are the -σ uncertainties. See text for details on the derivation of those quantities. Gaps in the time series are not unfrequent, and one should keep in mind that they are not indicated in this table. a A -day campaign measurement has also been done with a different antenna on the station s monument in October 9. b 4 additional points in May 9 made the positions at this station exploitable. 7

8 (λ <.8) lead to models featuring locked patches only underneath data points, right next to creeping patches. Besides being unphysical and resulting in very high reduced chi squares, such models are highly dependent on the data spatial distribution and must then be rejected. A smoothing too large (λ > ) tends to lead to a fault locked further at depth, and with a very smooth locked-creeping transition, which doesn t fit the data anymore (reduced χ > 3 on figure S4). Within the range of Laplacian weight.8 < λ <, the moment deficit accumulated each year remains within the uncertainties determined by the inversion. Moment deficit rate (Nm/yr) 7. x " continuous GPS " campaign GPS " leveling " total Moment deficit rate " of different data sets Laplacian weight (! parameter) Figure S4: Variation of the χ of the fit and of the moment deficit rate for different values of the weight attributed to the Laplacian in the inversion. The black curve shows the moment deficit accumulated every year as a function of the weight attributed to the Laplacian. The dashed black line and grey shaded area represent the rate of moment deficit with uncertainties derived in this study, i.e M = 6.6 ±.4 9 Nm/yr. The green, red and blue curves respectively represent the value of the χ of the fit to the continuous GPS, campaign and leveling data. S.4. Direction of extension of the Tibetan plateau Figure S shows the sensitivity of the long term velocity and the moment accumulation rate estimated in this study to the direction chosen for the extension of the Tibetan Plateau. The direction N98.E has been chosen because it is the one that affects the least the estimates of the long term velocities (it is the most perpendicular direction to those velocities, i.e. it is the direction onto which the sum of the projections of the East and West long term velocities 8

9 reaches a minimum). But there is no real reason to prevent this direction from varying by a few degrees from the N98.E azimuth. Figure S shows that even by changing this direction by, the final values of the parameters remain within their estimated uncertainties. Far velocity (mm/yr) v west v east 9 9 x 9 Moment deficit (Nm/yr) Azimuth of extension (N o E) Figure S: Impact of the azimuth selected for the extension of the Tibetan plateau on the long term East and West velocities (upper plot) and the moment deficit rate (lower plot). The solid thick lines represent the values of the parameters with respect the azimuth, the horizontal dashed lines and filled area of corresponding colors are the values with - σ uncertainties that we retained in this study (corresponding to an azimuth of N98.E): V e = 7.8 ±. mm/yr, V w =. ± mm/yr and M = 6.6 ±.4 9 Nm/yr. S.4.3 Recurrence time of large eartquakes Assuming that the moment deficit of M = 6.6 ±.4 9 Nm/yr computed in the main paper was released through earthquakes following a Gutenberg-Richter distribution up to a maximum magnitude, the recurrence time of those largest earthquakes (corresponding to those largest magnitudes) is plotted on figure S6(a). The black lines (solid, dashed and dotted) correspond to M = Nm/yr, with different percentages of this moment deficit being released seismically, while the grey surrounding lines show the extent corresponding to the uncertainties on M. This plot shows that earthquakes as large as the 9 Assam earthquake, whose moment magnitude is estimated at M w 8. (Ambraseys and Douglas, 4; Chen and Molnar, 977), could happen as often as once every 7 years within the borders of Nepal. As far as frequency is concerned, this would be the worst case 9

10 scenario where all the moment deficit accumulated was released seismically in earthquakes whose magnitude wouldn t exceed 8.. However, too many parameters remain unknown to make any accurate estimation on the return period of major earthquakes. Should the actual b-value of the seismicity distribution in Nepal slightly differ from, equation (8) shows that those estimates would be significantly affected. Another unknown parameter is the largest possible earthquake magnitude in Nepal which has a paramount effect, as shown on figure S6(b). Indeed if the seismicity on the MHT doesn t go beyond those M w 8. earthquakes, they would indeed have a period of return of about 7 years. But if we assume that the MHT can produce earthquakes up to M w 9., then the return period of M w 8. earthquakes would become of the order of 6 years. Recurrence period (yr)! =.! =.8! = Maximum M w of earthquakes (a) Recurrence time of the largest possible earthquakes in Nepal assuming a release of a proportion α of the accumulated moment by a seismicity following a Gutenberg-Richter distribution with b =. 9 T REC (M w = 8.) (yr) Maximum M w of earthquakes (b) Reccurence time of M w 8. earthquakes as a function of the largest possible earthquakes happening in Nepal, for a moment accumulation of M = Nm/yr released entirely seismically. Figure S6: Estimations on the recurrence time of earthquakes.

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